I’m assuming everyone who took high-school math class is familiar with Pythagoras and his legendary theorem, a^2 + b^2 = c^2. What fun…
But I wonder if people outside the scope of music theory know about his beautiful circle of fifths, my favorite circle. This one circle describes every relationship between the 12 tones in our scale. This. One. Picture. It is beyond amazing.
Anyone ever have a music theory class, where your teacher would say, “this is the rule, and don’t ask why, because nobody really knows why”? Well, I beg to differ. This representation of music tells you why, to almost any question about music theory that I can think of.
By looking at the outer circles, labelled with letters, it is clear how this illustrious circle got its name. There are 12 tones, with each one represented as a piece of the pie, and the relationship determining which order they go in, is simply “Add 5”. So you see, from F to C is five letters or more precisely, a fifth interval (7 semitones). From C to G, from G to D, on and on, the relationship is the same all the way around the circle. Inversely, it can be called the circle of fourths, because going counterclockwise, the relationship between each pie piece is a fourth interval, or four letters. Any inverted intervals always add up to 9 by the way, another unrelated piece of awesomeness in music theory (ie: a third inverted is always a sixth=9, therefore woohoo math!).
SO it here it goes. The Key of C (and A minor) has no sharps or flats. The next is G, which has one sharp. Then D with 2 sharps, and so on and so forth. Using this, you can see the relationship between keys. Because of this, composers can make quick observations about related keys, modulations, building harmony and chords, and essentially anything that you need to have a piece of music except your own creativity.
What I think is really neat, is that it also solidifies why everything in tonal music is built on 4ths and 5ths. A progression in any tonal song needs to go from I to IV to V and back to I for the listener to feel satisfied, or sometimes a different combination of those 3 chords. Other stuff is only added in for variety, basically. Why? Because math..look at that circle!
Why are the 4th and 5th scale of any called the sub-dominant and dominant (a.k.a. the most important other than the tonic)? Because Pythagoras knows everything!
Why are 4th and 5th intervals called “perfect” when no other interval other than unison gets that high distinction? Because they are perfect.
The relationship of all tones is essentially defined by this circle (which is perfect, like all circles) that is built on 4ths and 5ths. These 2 numbers seem to be magic when it comes to music, but not magic, just math.
God I just love it when everything makes sense!